Arcs in $\mathbb F_q^2$

Abstract

An arc is a subset of $\mathbb F_q^2$ which does not contain any collinear triples. Let $A(q,k)$ denote the number of arcs in $\mathbb F_q^2$ with cardinality $k$. This paper is primarily concerned with estimating the size of $A(q,k)$ when $k$ is relatively large, namely $k=q^t$ for some $t>0$. Trivial estimates tell us that \[ {q \choose k} \leq A(q,k) \leq {q^2 \choose k}. \] We show that the behaviour of $A(q,k)$ changes significantly close to $t=1/2$. Below this threshold an elementary argument is used to prove that the trivial upper bound above cannot be improved significantly. On the other hand, for $t \geq 1/2+δ$, we use the theory of hypergraph containers to get an improved upper bound \[ A(q,k) \leq {q^{2-t+2δ} \choose k}. \] This technique is also used to give an upper bound for the size of the largest arc in a random subset of $\mathbb F_q^2$ which holds with high probability. For example, we prove that a $p$-random subset $Q \subset \mathbb F_q^2$ with $q^{-3/2}<p<q^{-1}$ contains an arc of size $Ω(q^{1/2})$ with high probability. The result is optimal for this range of $p$. Finally, this optimal bound for arcs in random sets is used to prove a finite field analogue of a result of Balogh and Solymosi, with a better exponent: there exists a subset $P \subset \mathbb F_q^2$ which does not contain any collinear quadruples, but with the property that for every $P' \subset P$ with $|P'| \geq |P|^{3/4+o(1)}$, $P'$ contains a collinear triple.